Cantor Chocolate

Calculus Level 2

Jill takes a bar of chocolate:

She removes the middle third and gives it to her sister, as shown:

This is what Jill has left now

She then removes the middle thirds from the two remaining chocolate bars and gives them to her sister again.

Jill's share of the chocolate this time

If she continues to do this indefinitely with each of the pieces she has left, will her sister eventually receive the entirety of the chocolate bar?

No, Jill will be left with infinitesimally thin pieces Yes

3 solutions

R Mathe
Jun 15, 2018

The answer really depends on what you mean by ‘entire’ chocolate bar. Topologically speaking, she recieves all but a meagre subset. Geometrically speaking, she receieves all but a nullset. In both these senses she recieves ‘essentially’ the whole bar.

B D
Sep 2, 2018

The answer is No, Jill will be left with infinitesimally thin pieces.

Her sister gets 1/3 of the chocolate bar which is two times less then Jill. So Jill has more then her sister which is isn't the entirety chocolate bar.

Intouch Piticharoen
Jun 11, 2018

Although the sum of the Cantor Set converges to 1, it requires taking the limit to infinity.
We cannot actually perform an infinite amount of steps so there will always be a bit of chocolate left every iteration.

So if Jill's sister does this (assuming she has a infinitely fine microscope and cutting equipment), she'll never finish.

Assume the $n$ th takes $2^{-n}$ minute. Then after $1$ minute, the process is complete (even though the person doing the cutting has experienced an eternal sequence of cuts before this). It’s an idealised problem taking place in the ‘mathsverse’, not the universe.

R Mathe - 2 years, 12 months ago

Cantor Chocolate

3 solutions

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